INTRODUCTION TO THE MECHANICS OF SPACE ROBOTS SPACE TECHNOLOGY LIBRARY Published jointly by Microcosm Press and Springer
The Space Technology Library Editorial Board
Managing Editor: James R. Wertz, Microcosm, Inc., El Segundo, CA, USA; Editorial Board: Val A. Chobotov, Consultant on Space Hazards, Aerospace Corporation, Los Angeles, CA, USA; Michael L. DeLorenzo, Permanent Professor and Head, Dept. of Astronau- tics, U.S. Air Force Academy, Colorado Spring, CO, USA; Roland Doré, Professor and Director, International Space University, Stras- bourg, France; Robert B. Giffen, Professor Emeritus, U.S. Air Force Academy, Colorado Spring, CO, USA; Gwynne Gurevich, Space Exploration Technologies, Hawthorne, CA, USA; Wiley J. Larson, Professor, U.S. Air Force Academy, Colorado Spring, CO, USA; Tom Logsdon, Senior Member of Technical Staff, Space Division, Rockwell International, Downey, CA, USA; F. Landis Markley, Goddard Space Flight Center, NASA, Greenbelt, MD, USA; Robert G. Melton, Associate Professor of Aerospace Engineering, Pennsyl- vania State University, University Park, PA, USA; Keiken Ninomiya, Professor, Institute of Space & Astronautical Science, Sagamihara, Japan; Jehangir J. Pocha, Letchworth, Herts, UK; Frank J. Redd, Professor and Chair, Mechanical and Aerospace Engineer- ing Dept., Utah State University, Logan, UT, USA; Rex W. Ridenoure, Jet Microcosm, Inc., Torrance, CA, USA; Malcolm D. Shuster, Professor of Aerospace Engineering, Mechanics and Engineering Science, University of Florida, Gainesville, FL, USA; Gael Squibb, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA; Martin Sweeting, Professor of Satellite Engineering, University of Surrey, Guildford, UK
For further volumes: www.springer.com/series/6575 Giancarlo Genta
Introduction to the Mechanics of Space Robots Prof. Dr. Giancarlo Genta Department of Mechanical and Aerospace Engineering Politecnico di Torino Corso Duca degli Abruzzi 24 Torino 10129 Italy [email protected]
ISBN 978-94-007-1795-4 e-ISBN 978-94-007-1796-1 DOI 10.1007/978-94-007-1796-1 Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011935862
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Preface
This text started as a collection of notes of the lectures on Space Robotics given by the author to the students of the International Master on Space Exploration and De- velopment Systems (SEEDS). The aim of the course was the study of the automatic machines aimed to operate both autonomously and as a support to astronauts in space exploration and exploitation missions, with particular attention to the devices designed for planetary environment, including small planets, comets and asteroids. This material was then completed and made more systematic so that it can hope- fully be useful not only to the students of that course but also to those who have an interest in the wide and much interdisciplinary field of space robotics, and in particular in its mechanical aspects. The focus is drawn mainly on the mechanics of space robots: the author is well aware that, even in this specific field, it is far from being complete and that robots, like all mechatronic systems, are so integrated that no single aspect can be dealt sep- arately. Many important aspects are either dealt with only marginally or altogether left out. The very important topics of the control and the behavior of robots, for in- stance, are only marginally touched, even if their influence on the mechanical aspect to which this book is dedicated is not at all marginal. The structure of the book is so organized: • Chapter 1: a very short introductory overview of human and robotic space ex- ploration, stressing the need for man-machine cooperation in exploration. The various types of robotic missions in LEO, deep space and on planets and their basic requirements are shortly summarized. • Chapter 2 deals in a synthetic way with the main characteristics of the environ- ments space robots are facing and will face in the future. Since space environment is a specialized subject, dealt with in many books, this subject is only briefly sum- marized. • The configurations of robot arms and the basic kinematic and dynamic relation- ships needed for their design are described in Chap. 3. • Chapter 4 is devoted to the study of mobility on planetary surfaces, using differ- ent kind of supporting devices, like wheels, legs and aerodynamic or aerostatic devices.
vii viii Preface
• The basic characteristics of wheeled robots and vehicles are summarized in Chap. 5. The behavior of wheeled devices is studied in its various aspects, like longitudinal, lateral and suspension dynamics. The consequences of operating wheeled machines in the various environments are analyzed in some detail. The chapter is concluded by a description of the only vehicle that successfully carried humans on the surface of the Moon, the Apollo Lunar Roving Vehicle. • Vehicles and robots that use legs, tracks or other devices to move on a solid sur- face are described in Chap. 6. Since a great number of different architectures were proposed and sometimes even used in the past, not all the possible configurations are illustrated: the choice was based on the actual existing applications and on the perspectives of future use. • Chapter 7 is devoted to a short overview of the transducers used for actuation and sensing in space robots. • A short overview of the energy sources and storage devices that can be used for space robots is reported in Chap. 8. The book includes two appendices summarizing the theoretical formulations al- lowing to write mathematical models of space robots including a variety of mechan- ical components, such as arms, legs, etc. The author found it necessary to include them, since the participants to the course in Space Exploration and Development Systems have a much varied background and what may seem obvious to some stu- dents, could be difficult for other ones. In a similar way, some of the readers of this book may not be familiar with the concepts of analytical mechanics or dynamics of deformable bodies used in the text, mainly in Chaps. 3 and 5. The author is grateful to colleagues and students of the Mechanics Department and the Mechatronics Laboratory of the Politecnico di Torino for their suggestions, criticism and general exchange of ideas. Students, in particular postgraduate stu- dents, cooperated to this book with their thesis work and their questions, but mainly with their very presence that compelled me to clarify my own ideas and to work out all details. To all of them goes my gratitude. Last, but far from least, this book could not have been written without the support, encouragement and patience by my wife Franca—advisor, critic, editor, companion and best friend since 44 years.
A Note on the Illustrations I have made every effort to seek permission from the original copyright holders of the figures, and I apologize if there are cases where I have not been able to achieve my objective. This applies in particular to figures taken from the web, like Figs. 4.10, 4.35, 4.36, 4.40, 5.2, 6.1, 6.15, 6.17, 6.21, 6.23b, 6.24, 6.25a, 6.28, 6.31a, b, c, 7.18, 7.19 and 7.21. Torino, Italy Giancarlo Genta Contents
1 Introduction ...... 1 1.1 Robots in Space ...... 1 1.2 Humans and Robots ...... 3 1.3 Artificial Intelligence ...... 7 1.4 Missions for Robots and Manipulators ...... 11 1.4.1LowEarthOrbit(LEO)...... 12 1.4.2 Deep Space ...... 14 1.4.3 Planetary Surfaces ...... 14 1.5OpenProblems...... 16 1.5.1Control...... 17 1.5.2 Mechanics ...... 17 1.5.3 Transducers ...... 18 1.5.4Power...... 18 1.5.5Communications...... 18 2 Space and Planetary Environment ...... 21 2.1LowEarthOrbitEnvironment...... 21 2.2 Interplanetary Medium ...... 25 2.3InterstellarMedium...... 27 2.4 Lunar Environment ...... 29 2.5 Rocky Planets ...... 35 2.5.1Mars...... 35 2.5.2Mercury...... 40 2.5.3 Venus ...... 42 2.6 Giant Planets ...... 43 2.6.1Jupiter...... 45 2.6.2Saturn...... 46 2.6.3 Uranus ...... 48 2.6.4Neptune...... 50 2.7 Satellites of Giant Planets ...... 51 2.7.1Io...... 54
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2.7.2Europa...... 54 2.7.3Ganymede...... 55 2.7.4 Callisto ...... 55 2.7.5 Enceladus, Tethys, Dione, Rhea and Iapetus ...... 56 2.7.6Titan...... 56 2.7.7 Miranda, Ariel, Umbriel, Titania and Oberon ...... 58 2.7.8Triton...... 58 2.8SmallBodies...... 59 2.8.1MainBeltAsteroids...... 59 2.8.2KuiperBeltObjects...... 62 2.8.3TrojanAsteroids...... 63 2.8.4OtherAsteroids...... 64 2.8.5Comets...... 65 2.8.6 Gravitational Acceleration on the Surface ofNon-regularAsteroids...... 67 3 Manipulatory Devices ...... 73 3.1 Degrees of Freedom and Workspace ...... 73 3.2EndEffectors...... 77 3.3OrientationoftheEndEffector...... 79 3.4 Redundant Degrees of Freedom ...... 80 3.5 Arm Layout ...... 82 3.6 Position of a Rigid Body in Tridimensional Space ...... 83 3.7 Homogeneous Coordinates ...... 86 3.8 DenavitÐHartenberg Parameters ...... 87 3.9KinematicsoftheArm...... 90 3.10VelocityKinematics...... 100 3.11ForcesandMoments...... 102 3.12 Dynamics of Rigid Arms ...... 103 3.13LowLevelControl...... 114 3.13.1OpenLoopControl...... 115 3.13.2Closed-LoopControl...... 115 3.13.3 Model-Based Feedback Control ...... 123 3.13.4 Mixed Feedforward and Feedback Control ...... 124 3.14 Trajectory Generation ...... 125 3.15 Dynamics of Flexible Arms ...... 128 3.16HighLevelControl...... 145 3.17ParallelManipulators...... 146 4 Mobility on Planetary Surfaces ...... 153 4.1 Mobility ...... 153 4.2 VehicleÐGround Contact ...... 154 4.2.1ContactPressure...... 156 4.2.2Traction...... 162 4.3 Wheeled Locomotion ...... 168 4.3.1 Stiff Wheels Rolling on Stiff Ground ...... 168 Contents xi
4.3.2 Compliance of the Wheel and of the Ground ...... 171 4.3.3 Contact Between Rigid Wheel and Compliant Ground . . . 174 4.3.4 Contact Between Compliant Wheel and Rigid Ground . . . 180 4.3.5 Contact Between Compliant Wheel and Compliant Ground ...... 186 4.3.6 Tangential Forces: Elastic Wheels on Rigid Ground ....190 4.3.7 Tangential Forces: Rigid Wheel on Compliant Ground . . . 208 4.3.8 Tangential Forces: Compliant Wheel on Compliant Ground ...... 213 4.3.9 Tangential Forces: Empirical Models ...... 215 4.3.10 Dynamic Behavior of Tires ...... 218 4.3.11 Omni-Directional Wheels ...... 218 4.4 Tracks ...... 220 4.5 Legged Locomotion ...... 221 4.6 Fluidostatic Support ...... 224 4.7 Fluid-Dynamics Support ...... 226 4.8 Other Types of Support ...... 234 5 Wheeled Vehicles and Rovers ...... 235 5.1 Introduction ...... 235 5.2 Uncoupling of the Equations of Motion of Wheeled Vehicles . . . 237 5.3LongitudinalBehavior...... 240 5.3.1 Forces on the Ground ...... 240 5.3.2 Resistance to Motion ...... 242 5.3.3 Model of the Driveline ...... 244 5.3.4 Model Including the Longitudinal Slip ...... 248 5.3.5 Maximum Torque that Can Be Transferred to the Ground ...... 253 5.3.6 Maximum Performances Allowed by the Motors ...... 255 5.3.7 Energy Consumption at Constant Speed ...... 257 5.3.8 Acceleration ...... 258 5.3.9Braking...... 260 5.4LateralBehavior...... 266 5.4.1TrajectoryControl...... 266 5.4.2 Low-Speed or Kinematic Steering ...... 269 5.4.3 Ideal Steering ...... 273 5.4.4 GroundÐWheel Contact as a Non-holonomic Constraint . . 278 5.4.5 Model for High-Speed Cornering ...... 285 5.4.6 Linearized Model for High-Speed Cornering ...... 288 5.4.7SlipSteering...... 310 5.4.8ArticulatedSteering...... 314 5.4.9 Trajectory Definition ...... 326 5.4.10SteeringActivity...... 329 5.5 Suspension Dynamics ...... 329 5.5.1 Non Compliant Suspensions ...... 331 5.5.2 Elastic Suspensions ...... 338 xii Contents
5.5.3 Anti-dive and Anti-squat Designs ...... 347 5.5.4 Quarter-Car Models ...... 350 5.5.5 Bounce and Pitch Motions ...... 358 5.5.6 Wheelbase Filtering ...... 364 5.5.7RollMotions...... 366 5.5.8 Ground Excitation ...... 367 5.5.9EffectsofVibrationontheHumanBody...... 369 5.5.10 Concluding Remarks on Ride Comfort ...... 370 5.6 Coupled Longitudinal, Lateral and Suspension Models ...... 373 5.7 The Apollo LRV...... 375 5.7.1 Wheels and Tires ...... 375 5.7.2DriveandBrakeSystem...... 376 5.7.3 Suspensions ...... 377 5.7.4Steering...... 377 5.7.5PowerSystem...... 378 5.8 Conclusions on Wheeled Vehicles ...... 379 6 Non-wheeled Vehicles and Rovers ...... 381 6.1 Walking Machines ...... 381 6.1.1 General Layout ...... 381 6.1.2 Generation of Feet Trajectories ...... 385 6.1.3 Non-zoomorphic Configurations ...... 389 6.1.4GaitandLegCoordination...... 397 6.1.5 Equilibrium ...... 401 6.1.6 Biped and Humanoid Robots ...... 404 6.1.7 Conclusions ...... 408 6.2 Hybrid Machines with Wheels and Legs ...... 410 6.3 Hybrid Machines with Tracks and Legs ...... 414 6.4 Hopping Robots ...... 416 6.5Skis...... 422 6.6 Apodal Devices ...... 423 7 Actuators and Sensors ...... 427 7.1 Actuation of Space Robots ...... 427 7.2 Linear Actuators ...... 429 7.2.1 Performance Indices ...... 429 7.2.2 Hydraulic Cylinders ...... 433 7.2.3 Pneumatic Actuators ...... 436 7.2.4 Solenoid Actuators ...... 437 7.2.5MovingCoilActuators...... 443 7.2.6 Piezoelectric Actuators ...... 445 7.3RotaryActuators...... 450 7.3.1ElectricMotors...... 450 7.3.2 Hydraulic and Pneumatic Motors ...... 458 7.3.3InternalCombustionEngines...... 459 Contents xiii
7.4 Mechanical Transmissions ...... 462 7.4.1FromRotarytoRotaryMotion...... 462 7.4.2 From Rotary to Linear Motion ...... 468 7.5HydraulicTransmissions...... 471 7.6 Sensors ...... 475 7.6.1 Exteroceptors ...... 476 7.6.2 Proprioceptors ...... 478 8 Power Systems ...... 483 8.1SolarEnergy...... 484 8.1.1 Photovoltaic Generators ...... 484 8.1.2 Solar-Thermal Generators ...... 487 8.2NuclearPower...... 487 8.2.1 Fission Reactors ...... 488 8.2.2 Radioisotope Generators ...... 489 8.2.3RadioisotopeHeatingUnits(RHUs)...... 492 8.3ChemicalPower(Combustion)...... 492 8.3.1ThermalEngines...... 494 8.3.2 Fuel Cells ...... 494 8.4 Electrochemical Batteries ...... 496 8.4.1PrimaryBatteries...... 496 8.4.2 Secondary (Rechargeable) Batteries ...... 498 8.5OtherEnergyStorageDevices...... 502 8.5.1 Supercapacitors ...... 502 8.5.2 Flywheels ...... 503 Appendix A Equations of Motion in the Configuration and State Spaces ...... 505 A.1 Discrete Linear Systems ...... 505 A.1.1 Configuration Space ...... 505 A.1.2 State Space ...... 507 A.1.3FreeMotion...... 509 A.1.4ConservativeNaturalSystems...... 511 A.1.5 Properties of the Eigenvectors ...... 512 A.1.6 Uncoupling of the Equations of Motion ...... 513 A.1.7 Natural Nonconservative Systems ...... 515 A.1.8 Systems with Singular Mass Matrix ...... 518 A.1.9 Conservative Gyroscopic Systems ...... 519 A.1.10 General Dynamic Systems ...... 520 A.1.11 Closed Form Solution of the Forced Response ...... 522 A.1.12 Modal Transformation of General Linear Dynamic Systems...... 523 A.2 Nonlinear Dynamic Systems ...... 523 A.3 Lagrange Equations in the Configuration and State Space .....525 A.4 Lagrange Equations for Systems with Constraints ...... 528 A.4.1 Holonomic Constraints ...... 529 A.4.2 Non-holonomic Constraints ...... 531 xiv Contents
A.5 Hamilton Equations in the Phase Space ...... 532 A.6 Lagrange Equations in Terms of Pseudo-Coordinates ...... 533 A.7MotionofaRigidBody...... 536 A.7.1 Generalized Coordinates ...... 536 A.7.2 Equations of Motion—Lagrangian Approach ...... 538 A.7.3 Equations of Motion Using Pseudo-Coordinates ...... 539 A.8 Multibody Modeling ...... 541 Appendix B Equations of Motion for Continuous Systems ...... 545 B.1 General Considerations ...... 545 B.2Beams...... 547 B.2.1 General Considerations ...... 547 B.2.2FlexuralVibrationsofStraightBeams...... 548 B.2.3 Effect of Shear Deformation ...... 559 B.3 Discretization of Continuous Systems: The FEM ...... 563 B.3.1ElementCharacterization...... 563 B.3.2 Timoshenko Beam Element ...... 566 B.3.3MassandSpringElements...... 573 B.3.4AssemblingtheStructure...... 574 B.3.5ConstrainingtheStructure...... 575 B.3.6DampingMatrices...... 576 B.4 Reduction of the Number of Degrees of Freedom ...... 577 B.4.1 Static Reduction ...... 578 B.4.2 Guyan Reduction ...... 579 B.4.3 Component-Mode Synthesis ...... 580 References ...... 585 Robotics...... 585 Terramechanics and Dynamics of Wheeled and Legged Vehicles...... 588 Index ...... 589 Symbols and Acronyms
Symbols a length of the contact area; acceleration distance between center of mass and front axle a acceleration vector b width of contact area; distance between center of mass and rear axle; wingspan c cohesive bearing strength; viscous damping coefficient; wing chord ccr critical damping copt optimal damping d soil deformation; diameter d direct piezoelectric matrix di second DH parameter: offsett e energy e error ei unit vector of the ith axis f friction coefficient; rolling coefficient f0 rolling coefficient at zero speed fr rollover factor fs sliding factor g gravitational acceleration g gravitational acceleration vector h sinking in the ground hc convection coefficient √ i grade of the road; imaginary unit (i = −1); current it transversal grade of the road k stiffness; modulus of soil deformation kc cohesive modulus kφ frictional modulus l length of the arm; wheelbase li third DH parameter: length m mass
xv xvi Symbols and Acronyms me equivalent mass ms sprung mass mu unsprung mass p pressure p generalized momenta p, q, r angular velocities in the xyz frame ps bearing capacity of the soil with no sinking p0 bearing capacity of the soil q eigenfunction q vector of the generalized coordinates; eigenvector r radius r vector s laplace variable t time; track; pneumatic trail; thickness u displacement u displacement vector u, v, w velocities in the xyz frame v volume vg velocity of the ground due to slip z sinking; number of teeth xyz body-fixed reference frame x coordinate vector z state vector A area A dynamic matrix in the state space Br magnetic remanence B input gain matrix C cornering stiffness; capacitance CD drag coefficient Cf force coefficient CL lift coefficient CS side force coefficient Cγ camber stiffness Cσ longitudinal force coefficient C damping matrix; output gain matrix D aerodynamic drag; displacement D direct link matrix; dynamic matrix in the configuration space E Young’s modulus; modulus of deformation (soil); aerodynamic efficiency E stiffness matrix of the material F force F force vector Fn normal force Fr Froude number Ft tangential force Symbols and Acronyms xvii
G shear modulus; gravitational constant G gyroscopic matrix Hc coercitive magnetic field H Hamiltonian function H circulatory matrix I identity matrix; inertia matrix J moment of inertia J Jacobian matrix K stiffness KB back EMF constant KT torque constant K stiffness matrix; matrix of the control gains Kd derivative gains matrix Ki integrative gains matrix Kp proportional gains matrix L reference length; aerodynamic lift L Lagrangian function M mass M mass matrix; moment M molecular mass Ma Mach number N number of turns Nu Nusselt number N matrix of the shape functions P power Q flow R radius of the wheel (unloaded); radius of the trajectory; universal gas constant; resistance to motion; electric resistance R reluctance R rotation matrix Rc radius of the trajectory (low speed conditions) Re Reynolds number Re effective rolling radius Rl radius under load S first order mass moment; aerodynamic side force; reference surface T temperature; torque T torque vector; homogeneous transformation matrix T kinetic energy U potential energy V vehicle speed; volume; voltage V velocity vector Vf velocity of the foot relative to the body Vr velocity relative to the atmosphere Vs velocity of sound VB back electromotive force xviii Symbols and Acronyms
W work XYZ inertial frame α sideslip angle; grade angle of the road; angle of attack αi fourth DH parameter: twist αt transversal grade angle of the road β sideslip angle of the vehicle; duty factor γ camber angle; inclination angle δ steering angle; aerodynamic sideslip angle; resistivity δc steering angle (low speed steering) δL virtual work δθ virtual displacement strain; deformation of the soil strain vector f efficiency of the brake η efficiency; modal coordinate ηi modal coordinates vector θ pitch angle; thermal resistance θi first DH parameter: rotation angle θ vector of the generalized coordinates at the joints λ thermal conductivity μ dynamic viscosity; traction coefficient μ0 magnetic permeability of vacuum μ∗ friction coefficient μr relative magnetic permeability μx longitudinal force coefficient
μxp longitudinal traction coefficient
μxs sliding longitudinal traction coefficient μy cornering force coefficient
μyp lateral traction coefficient
μys sliding lateral traction coefficient ν Poisson’s ratio; kinematic viscosity ρ density σ normal pressure; stress; longitudinal slip σ stress vector τ shear stress; transmission ratio; time delay; nondimensional time φ roll angle; friction angle (φ = atan(μ)) χ torsional stiffness ψ yaw angle ω frequency; circular frequency ωn natural frequency rotational damping coefficient h increase of sinking Π torsional stiffness of the tires of an axle matrix of the eigenvectors Ω angular velocity Symbols and Acronyms xix